Here, u is a function of two variables x and t, and the subscripts denote partial derivatives. We will assume that c is a fixed constant. Given an initial condition

(2)

we would like to find a function of two variables that satisfies both the transport equation (1) and the initial condition (2).

This equation can be used to model pollution (Lehn and Scherer , undated), dye dispersion (Roychoudhury , undated), or even traffic flow (Jungel , 2002), with u representing the density of the pollutant (or dye or traffic, respectively) at position x and time t. For a discussion of the physical model, see Knobel (2000). For a discussion of the more general transport equation and its solutions, see Cooper (1998). For discussion and simulation of more general conservation laws, including shock wave phenomena, see Sarra (2003).